July  2022, 27(7): 4097-4119. doi: 10.3934/dcdsb.2021219

The impact of time delay and angiogenesis in a tumor model

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

*Corresponding author: Huijuan Song

Received  March 2021 Revised  June 2021 Published  July 2022 Early access  September 2021

We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to $ \alpha $, and a parameter $ \mu $ is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution $ \left(\sigma_{*}, p_{*}, R_{*}\right) $ for all positive $ \alpha $, $ \mu $. Then a threshold value $ \mu_\ast $ is found such that the radially symmetric stationary solution is linearly stable if $ \mu<\mu_\ast $ and linearly unstable if $ \mu>\mu_\ast $. Our results indicate that the increase of the angiogenesis parameter $ \alpha $ would result in the reduction of the threshold value $ \mu_\ast $, adding the time delay would not alter the threshold value $ \mu_\ast $, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter $ \mu $ is, the greater impact of time delay would have on the size of the stationary tumor.

Citation: Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4097-4119. doi: 10.3934/dcdsb.2021219
References:
[1]

R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[2]

H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.

[3]

H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.  doi: 10.1016/0025-5564(94)00117-3.

[4]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.

[5]

S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.  doi: 10.1007/s002850100130.

[6]

S. Cui and S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523-541.  doi: 10.1016/j.jmaa.2007.02.047.

[7]

S. Cui and Y. Zhuang, Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.  doi: 10.1016/j.jmaa.2018.08.022.

[8]

M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206. 

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U. Foryś and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Model., 37 (2003), 1201-1209. 

[10]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.  doi: 10.1007/s002850050149.

[11]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.  doi: 10.1016/j.jde.2005.09.008.

[12]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330.  doi: 10.1007/s00205-005-0408-z.

[13]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[14]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634.  doi: 10.1090/S0002-9947-00-02715-X.

[15]

A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.  doi: 10.1142/S0218202507002467.

[16]

A. Friedman and K-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.  doi: 10.1016/j.jde.2015.08.032.

[17]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[18]

H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242.  doi: 10.1016/S0022-5193(76)80054-9.

[19]

Y. HuangZ. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.  doi: 10.1016/j.nonrwa.2016.12.003.

[20]

Y. HuangZ. Zhang and B. Hu, Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Contin. Dyn. Syst., 39 (2019), 2473-2510.  doi: 10.3934/dcds.2019105.

[21]

Y. HuangZ. Zhang and B. Hu, Linear stability for a free-boundary tumor model with a periodic supply of external nutrients, Math. Methods Appl. Sci., 42 (2019), 1039-1054.  doi: 10.1002/mma.5412.

[22]

Y. HuangZ. Zhang and B. Hu, Asymptotic stability for a free boundary tumor model with angiogenesis, J. Differential Equations, 270 (2021), 961-993.  doi: 10.1016/j.jde.2020.08.050.

[23]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.

[24]

S. XuQ. Zhou and M. Bai, Qualitative analysis of a time-delayed free boundary problem for tumor growth under the action of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 4187-4198.  doi: 10.1002/mma.3357.

[25]

S. Xu and Z. Feng, Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation, J. Math. Anal. Appl., 374 (2011), 178-186.  doi: 10.1016/j.jmaa.2010.08.043.

[26]

S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal. Real World Appl., 11 (2010), 401-406.  doi: 10.1016/j.nonrwa.2008.11.002.

[27]

S. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal. Real World Appl., 51 (2020), 103005. doi: 10.1016/j.nonrwa.2019.103005.

[28]

X. Zhao and B. Hu, The impact of time delay in a tumor model, Nonlinear Anal. Real World Appl., 51 (2020), 103015. doi: 10.1016/j.nonrwa.2019.103015.

[29]

X. Zhao and B. Hu, Symmetry-breaking bifurcation for a free-boundary tumor model with time delay, J. Differential Equations, 269 (2020), 1829-1862.  doi: 10.1016/j.jde.2020.01.022.

[30]

F. ZhouJ. Escher and S. Cui, Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumor, J. Math. Anal. Appl., 337 (2008), 443-457.  doi: 10.1016/j.jmaa.2007.03.107.

[31]

F. Zhou and S. Cui, Bifurcations for a multidimensional free boundary problem modeling the growth of tumor cord, Nonlinear Anal. Real World Appl., 10 (2009), 2990-3001.  doi: 10.1016/j.nonrwa.2008.10.004.

[32]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.  doi: 10.1016/j.jde.2018.03.005.

[33]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, Acta Appl. Math., 161 (2019), 153-169.  doi: 10.1007/s10440-018-0208-8.

show all references

References:
[1]

R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.  doi: 10.1016/j.bulm.2003.11.002.

[2]

H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117.  doi: 10.1016/S0025-5564(97)00023-0.

[3]

H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.  doi: 10.1016/0025-5564(94)00117-3.

[4]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.

[5]

S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.  doi: 10.1007/s002850100130.

[6]

S. Cui and S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523-541.  doi: 10.1016/j.jmaa.2007.02.047.

[7]

S. Cui and Y. Zhuang, Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.  doi: 10.1016/j.jmaa.2018.08.022.

[8]

M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206. 

[9]

U. Foryś and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Model., 37 (2003), 1201-1209. 

[10]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.  doi: 10.1007/s002850050149.

[11]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.  doi: 10.1016/j.jde.2005.09.008.

[12]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330.  doi: 10.1007/s00205-005-0408-z.

[13]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[14]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634.  doi: 10.1090/S0002-9947-00-02715-X.

[15]

A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.  doi: 10.1142/S0218202507002467.

[16]

A. Friedman and K-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.  doi: 10.1016/j.jde.2015.08.032.

[17]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[18]

H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242.  doi: 10.1016/S0022-5193(76)80054-9.

[19]

Y. HuangZ. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.  doi: 10.1016/j.nonrwa.2016.12.003.

[20]

Y. HuangZ. Zhang and B. Hu, Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Contin. Dyn. Syst., 39 (2019), 2473-2510.  doi: 10.3934/dcds.2019105.

[21]

Y. HuangZ. Zhang and B. Hu, Linear stability for a free-boundary tumor model with a periodic supply of external nutrients, Math. Methods Appl. Sci., 42 (2019), 1039-1054.  doi: 10.1002/mma.5412.

[22]

Y. HuangZ. Zhang and B. Hu, Asymptotic stability for a free boundary tumor model with angiogenesis, J. Differential Equations, 270 (2021), 961-993.  doi: 10.1016/j.jde.2020.08.050.

[23]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.

[24]

S. XuQ. Zhou and M. Bai, Qualitative analysis of a time-delayed free boundary problem for tumor growth under the action of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 4187-4198.  doi: 10.1002/mma.3357.

[25]

S. Xu and Z. Feng, Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation, J. Math. Anal. Appl., 374 (2011), 178-186.  doi: 10.1016/j.jmaa.2010.08.043.

[26]

S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal. Real World Appl., 11 (2010), 401-406.  doi: 10.1016/j.nonrwa.2008.11.002.

[27]

S. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal. Real World Appl., 51 (2020), 103005. doi: 10.1016/j.nonrwa.2019.103005.

[28]

X. Zhao and B. Hu, The impact of time delay in a tumor model, Nonlinear Anal. Real World Appl., 51 (2020), 103015. doi: 10.1016/j.nonrwa.2019.103015.

[29]

X. Zhao and B. Hu, Symmetry-breaking bifurcation for a free-boundary tumor model with time delay, J. Differential Equations, 269 (2020), 1829-1862.  doi: 10.1016/j.jde.2020.01.022.

[30]

F. ZhouJ. Escher and S. Cui, Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumor, J. Math. Anal. Appl., 337 (2008), 443-457.  doi: 10.1016/j.jmaa.2007.03.107.

[31]

F. Zhou and S. Cui, Bifurcations for a multidimensional free boundary problem modeling the growth of tumor cord, Nonlinear Anal. Real World Appl., 10 (2009), 2990-3001.  doi: 10.1016/j.nonrwa.2008.10.004.

[32]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.  doi: 10.1016/j.jde.2018.03.005.

[33]

Y. Zhuang and S. Cui, Analysis of a free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, Acta Appl. Math., 161 (2019), 153-169.  doi: 10.1007/s10440-018-0208-8.

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